Special Series 1: Sum of first n natural numbers
The result of this series is given below:
1+ 2 + 3 + 4 + …. + n = n (n + 1) / 2
Proof:
Let Sn = 1 + 2 + 3 + 4 + … + n
We can see that this is an Arithmetic Progression with the first term (a) = 1 and common difference (d) =1 and there are n term
So, Sum of n terms = n/2 (2 x a + (n – 1) x d)
Putting the values for this series we will get
Sn = n/2(2 x 1 + (n – 1) x 1)
Sn = n/2(2 + n – 1)
Sn = n(n + 1)/2
Hence Proved.
Example on Sum of first n natural numbers
Question. Find the sum of the following series 3 + 4 + 5 —- + 25?
Solution:
Let Sn = 3+ 4 + 5 — + 25
Now we can also write it like this
Sn + 1 + 2 = 1 + 2 + 3 + 4 —- + 25
Clearly now it is the sum of first 25 natural number we can be written like this
Sn + 1+ 2 = 25 (25 + 1) / 2
Sn = 325 – 1 – 2
Sn = 322
Special Series in Maths – Sequences and Series | Class 11 Maths
Special Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 + 7 + ….. The series special in some way or the other is called a special series. The following are the three types of special series.
- 1 + 2 + 3 +… + n (sum of first n natural numbers)
- 12 + 22 + 32 +… + n2 (sum of squares of the first n natural numbers)
- 13 + 23 + 33 +… + n3 (sum of cubes of the first n natural numbers)
In this article, we will read about special series in maths and how to get the formula for all these series. We will also see solved examples and practice problems on special series.
Table of Content
- What are Special Series?
- Special Series 1: Sum of first n natural numbers
- Example on Sum of first n natural numbers
- Special Series 2: Sum of squares of the first n natural numbers
- Examples on Sum of squares of the first n natural numbers
- Special Series 3: Sum of cubes of the first n natural numbers
- Example on Sum of cubes of the first n natural numbers
- Practice Problems on Special Series